New Formulae of Squares of Some Jacobi Polynomials Via Hypergeometric Functions

Abstract

In this article, a new formula expressing explicitly the squares of Jacobi polynomials of certain parameters in terms of Jacobi polynomials of arbitrary parameters is derived. The derived formula is given in terms of ceratin terminating hypergeometric function of the type $_4F_3(1)$. In some cases, this $_4F_3(1)$ can be reduced by using some well-known reduction formulae in literature such as Watson's and Pfa-Saalschutz's identities. In some other cases, this $_4F_3(1)$ can be reduced by means of symbolic computation, and in particular Zeilberger's, Petkovsek's and van Hoeij's algorithms. Hence, some new squares formulae for Jacobi polynomials of special parameters can be deduced in reduced forms which are free of any hypergeometric functions.